assignment_4_corrected[1]

assignment_4_corrected[1] - Math 400 Assignment #4...

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Unformatted text preview: Math 400 Assignment #4 (corrected) 1. Due: May 11, 2010 Find approximate numerical solutions on the interval [ 0, 2] for the differential equation dx = t - 2 x , with initial condition t0 = 0, x0 = 1 dt Use Euler's method, the modified Euler method, and a fourth order Runge - Kutta method, each with step sizes h = 0.2, 0.1, and 0.05. 5 1 Show that x = x ( t ) = e −2 t + ( 2t -1) is the exact solution to this initial value problem. 4 4 For each approximate solution provide a table, a graph, and the relative error at t = 1 and t = 2. 2. Find approximate numerical solutions on the interval [ 0,1.2] for the differential equation d2x ′ + 4π 2 x = 0 , with initial condition t0 = 0, x0 = 1, x0 = 0 2 dt Use Euler's method with step sizes h = 0.2, 0.1, and 0.05. Then show that x = x ( t ) = cos 2π t is the exact solution to this initial value problem. For each approximate solution provide a table, a graph, and the relative error at t = .5 and t = 1. 3. 4. 1 = − cos x , correct to seven 1 + e −2 x decimal places. (Suggestion: first graph the functions to see about where the roots lie.) Find the two smallest positive roots of the equation The system, 3x 2 − y 2 = 0 has a solution not too far from the point (1, 1). Find this root two 3 xy 2 − x 2 = 1 ways. First using the two-dimensional version of Newton’s method, and second by eliminating y, solving for x using the one-dimensional Newton’s method, and substituting to find y. ...
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